Hurwitz's Irrational Number Theorem

As Lagrange showed, any Irrational Number $\alpha$ has an infinity of rational approximations $p/q$ which satisfy

$$\left\vert{\alpha-{p\over q}}\right\vert < {1\over \sqrt{5}\,q^2}.$$ (1)

Similarly, if $\alpha\not={\textstyle{1\over 2}}(1+\sqrt{5}\,)$,

$$\left\vert{\alpha-{p\over q}}\right\vert < {1\over \sqrt{8}\,q^2},$$ (2)

and if $\alpha\not={\textstyle{1\over 2}}(1+\sqrt{5}\,)\not=\sqrt{2}$,

$$\left\vert{\alpha-{p\over q}}\right\vert < {5\over \sqrt{221}} {1\over q^2}.$$ (3)

In general, even tighter bounds of the form

$$\left\vert{\alpha-{p\over q}}\right\vert<{1\over L_n q^2}$$ (4)

can be obtained for the best rational approximation possible for an arbitrary irrational number $\alpha$, where the $L_n$ are called Lagrange Numbers and get steadily larger for each ``bad'' set of irrational numbers which is excluded.

See also Hurwitz's Irrational Number Theorem, Liouville's Rational Approximation Theorem, Liouville-Roth Constant, Markov Number, Roth's Theorem, Segre's Theorem, Thue-Siegel-Roth Theorem

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 40, 1987.

Chandrasekharan, K. An Introduction to Analytic Number Theory. Berlin: Springer-Verlag, p. 23, 1968.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187-189, 1996.